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06.11.2019 | Ausgabe 1/2020 Open Access

Journal of Combinatorial Optimization 1/2020

On the König deficiency of zero-reducible graphs

Journal of Combinatorial Optimization > Ausgabe 1/2020
Miklós Bartha, Miklós Krész
Wichtige Hinweise
The second author acknowledges the European Commission for funding the InnoRenew CoE project (Grant Agreement #739574) under the Horizon2020 Widespread-Teaming program and the Republic of Slovenia (Investment funding of the Republic of Slovenia and the European Union of the European regional Development Fund). Miklós Krész is also grateful for the support of the EU-funded Hungarian grant EFOP-3.6.2-16-2017-00015 and the Slovenian ARRS grant J1-9110.

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A confluent and terminating reduction system is introduced for graphs, which preserves the number of their perfect matchings. A union-find algorithm is presented to carry out reduction in almost linear time. The König property is investigated in the context of reduction by introducing the König deficiency of a graph G as the difference between the vertex covering number and the matching number of G. It is shown that the problem of finding the König deficiency of a graph is NP-complete even if we know that the graph reduces to the empty graph. Finally, the König deficiency of graphs G having a vertex v such that \(G-v\) has a unique perfect matching is studied in connection with reduction.

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